If we use the model
we have to estimate IJ means and 
(a total of IJ+1
parameters) using only IJ observations! Since we can't estimate all
of our parameters, we will change models (slightly),
where 
is the effect of factor A and 
is the effect
of factor B. Now we only have to estimate I+J+1 parameters, which
is now possible. (Actually, we also assume
which leaves us with only
I+J-1 parameters to estimate.)
A slightly more general additive model is
where 
are the number of replications at each combination
of factor A and factor B levels.
NOTE:\ When k is small, especially when k=1, we are forced to use the additive model. There will be more about this in Section 10.2.2.
The ANOVA table for the additive model is given by
The relevant null hypotheses are
and are tested by 
and 
, respectively. In words, these
hypotheses are
EXAMPLE:\ In a study of automobile traffic and air pollution, air samples taken at four different times and at five different locations were analyzed to obtain the amount of particulate matter present in the air. Is there any difference in true average amount of particulate matter present in the air due either to different sampling times or to different locations?
Notice that in this case, both and are significantly
greater than one. Thus, there is an effect due both to time and
location.
When the additive model holds, there is no interaction between factors A and B. In other words, the effect of factor A is the same no matter what the level of factor B is. When the additive model doesn't hold, we have to go to a model which allows A and B to interact.